SOLUTION: Factor Completely. 81t^2 + 4 - 36t = Thank you!!

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Question 322786: Factor Completely.
81t^2 + 4 - 36t =
Thank you!!
Found 4 solutions by checkley77, jim_thompson5910, stanbon, elvira:
Answer by checkley77(12844)   (Show Source): You can put this solution on YOUR website!
81t^2 + 4 - 36t
81T^2-36T+4
(9T-2)^2 ans.
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Start with the given expression.


Sort the terms in descending powers (ie sort the exponents in descending order)


Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .


Now multiply the first coefficient by the last term to get .


Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?


To find these two numbers, we need to list all of the factors of (the previous product).


Factors of :
1,2,3,4,6,9,12,18,27,36,54,81,108,162,324
-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-81,-108,-162,-324


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to .
1*324 = 324
2*162 = 324
3*108 = 324
4*81 = 324
6*54 = 324
9*36 = 324
12*27 = 324
18*18 = 324
(-1)*(-324) = 324
(-2)*(-162) = 324
(-3)*(-108) = 324
(-4)*(-81) = 324
(-6)*(-54) = 324
(-9)*(-36) = 324
(-12)*(-27) = 324
(-18)*(-18) = 324

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :


First NumberSecond NumberSum
13241+324=325
21622+162=164
31083+108=111
4814+81=85
6546+54=60
9369+36=45
122712+27=39
181818+18=36
-1-324-1+(-324)=-325
-2-162-2+(-162)=-164
-3-108-3+(-108)=-111
-4-81-4+(-81)=-85
-6-54-6+(-54)=-60
-9-36-9+(-36)=-45
-12-27-12+(-27)=-39
-18-18-18+(-18)=-36



From the table, we can see that the two numbers and add to (the middle coefficient).


So the two numbers and both multiply to and add to


Now replace the middle term with . Remember, and add to . So this shows us that .


Replace the second term with .


Group the terms into two pairs.


Factor out the GCF from the first group.


Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


Combine like terms. Or factor out the common term


Condense the terms.


===============================================================


Answer:


So factors to .


In other words, .


Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
Factor Completely.
81t^2 + 4 - 36t =
-----
Rearrange:
81t^2 - 36t + 4
---
Factor:
(9t-2)^2
===============
Cheers,
Stan H.
Answer by elvira(1)   (Show Source): You can put this solution on YOUR website!
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234235556=
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